The following problem is similar in spirit to some which were studied by Archimedes and others. Solve it using integral calculus: Let Ah be the closed region in the coordinate plane defined by the vertical lines 1 = x and x = h (where h > 1), the x-axis, and the hyperbola y = ((x^2) − 1)^1/2, and let Bh be the corresponding region defined by the vertical lines 0 = x and x = h (where h > 0), the x-axis, and the hyperbola’s asymptote y = x. Next, let Ph and Qh be the solids of revolution obtained by rotating Ah and Bh (respectively) about the x-axis. Compute the ratio |Vol (Ph)|/|Vol (Qh)|
The following problem is similar in spirit to some which were studied by Archimedes and others. Solve it using integral calculus: Let Ah be the closed region in the coordinate plane defined by the vertical lines 1 = x and x = h (where h > 1), the x-axis, and the hyperbola y = ((x^2) − 1)^1/2, and let Bh be the corresponding region defined by the vertical lines 0 = x and x = h (where h > 0), the x-axis, and the hyperbola’s asymptote y = x. Next, let Ph and Qh be the solids of revolution obtained by rotating Ah and Bh (respectively) about the x-axis. Compute the ratio |Vol (Ph)|/|Vol (Qh)|
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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The following problem is similar in spirit to some which were studied
by Archimedes and others. Solve it using
in the coordinate plane defined by the vertical lines 1 = x and x = h (where h > 1), the
x-axis, and the hyperbola y =
((x^2) − 1)^1/2, and let Bh be the corresponding region defined
by the vertical lines 0 = x and x = h (where h > 0), the x-axis, and the hyperbola’s
asymptote y = x. Next, let Ph and Qh be the solids of revolution obtained by rotating Ah
and Bh (respectively) about the x-axis. Compute the ratio
|Vol (Ph)|/|Vol (Qh)|
.
[Hint: Draw a picture to make the problem more transparent.]
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